It is known that the resonant frequency of crystal reference elements varies over temperature. FIG. 1a illustrates the resonant frequency variation of an AT-cut crystal (expressed in parts per million (PPM)) over temperature. Those skilled in the art will appreciate that the crystal performance curve illustrated in FIG. 1a may be expressed mathematically by the following equation: EQU f(T)=fo+a.sub.1 (T-To)+a.sub.2 (T-To).sup.2 +a.sub.3 (T-To).sup.3
where
T is the temperature PA1 f(T) is the resonant frequency of the crystal at temperature T, and PA1 fo is the resonant frequency of the crystal at temperature To. As can be seen, the performance over a temperature range of -5.degree. C. to 60.degree. C. is substantially linear, and is centered around an inflection point To at 25.degree. C.
As is known, the first, second and third order coefficients a.sub.1, a.sub.2, and a.sub.3 of equation (1) vary such that each crystal must be separately characterized to determine its performance over temperature. The effect of variations of the first order coefficient a.sub.1 causes the curve of FIG. 1a to be rotated about the center point To. Accordingly, it is customary to sort or "grade" crystals into one or more groups having different operational ranges over temperature based on variations of the first order coefficient a.sub.1. One such selection is illustrated in FIG. 1b. As can be seen, the variations of the first order coefficient of equation 1 have been separated into three groups: 5-10 PPM; 10-15 PPM; and 15-20 PPM, each group having 5 PPM range.
When designing an oscillator circuit, it is customary to include a compensation circuit which maintains a constant oscillator output frequency within a specified temperature range. In a manufacturing environment, the compensation circuit must be manually adjusted (or optimized) depending upon the "grading" of the crystal element. This practice is both laborious and highly susceptible to human error. Improper adjustments to the compensation circuit due to errors in crystal grading process or in the optimization of the compensation circuit may lead to erratic or degraded output frequency stability of the oscillator circuit as the ambient temperature varies.
Additionally, this technique does not account for variations caused by the second and/or the third order temperature coefficients a.sub.2 and a.sub.3, the effects of which may be significant in hot or cold temperature regions (i.e., below -10.degree. C. and above +65.degree. C.).
Accordingly, a need exists for a crystal compensation process that is immune to the human errors typified by current manufacturing processes and covers a wider temperature compensation range.